Numerical Solution of Micropolar Casson Fluid Behaviour on Steady MHD Natural Convective Flow about A Solid Sphere

5, Issue 1 (218) 55-66 Journal of Advanced Research in Fluid Mechanics and Thermal Sciences Journal homepage: www.akademiabaru.com/arfmts.html ISSN: 2289-7879 Numerical Solution of Micropolar Casson Fluid Behaviour on Steady MHD Natural Convective Flow about A Solid Sphere Open Access Hamzeh Taha Alkasasbeh 1, 1 Department of Mathematics, Faculty of Science, Ajloun National University, P.O. Box 43, Ajloun 2681, Jordan ARTICLE INFO Article history: Received 15 April 218 Received in revised form 25 July 218 Accepted 28 July 218 Available online 26 September 218 Keywords: Magnetohydrodynamic (MHD), microploar Casson fluid, numerical solution, sold sphere ABSTRACT In this study, steady laminar 2D (MHD) natural convection flow suspended micropolar Casson fluid over a solid sphere is investigated. The governing partial differential equations are transformed into dimensionless form and then solved numerically using an implicit finite difference scheme known as Keller-box method. The behaviors of different parameters namely, Casson fluid parameter, magnetic parameter and micropolar parameter on the local Nusselt number and the local skin friction coefficient, as well as the temperature, velocity and angular velocity have been examined graphically. From graphical results, it is found that as Casson parameter increases the local Nusselt number and angular velocity profiles increase and local skin friction coefficient, temperature and velocity profiles decrease. Copyright 218 PENERBIT AKADEMIA BARU - All rights reserved 1. Introduction The micropolar fluid, a subclass of microfluids which exhibit certain microscopic effects arising from the local structure and micro-motions of the fluid elements. Eringen [1] was the first who introduce the theory of micropolar fluids describe the micro-rotation and micro-inertia effects. In the last few years, many researchers have investigated and reported these effects on heat transfer flow problems in fluid dynamics. Agarwal et al., [2] studied the micropolar fluid flow with heat transfer over a porous stationary wall. Rebhi et al., [3] considered the natural convection flow of a micropolar fluid over a moving vertical porous plate with chemical reaction and absorption or heat generation effects. Usman et al., [4] investigated unsteady free convection flow and mass transfer in a micropolar fluid from vertical permeable surface with variable suction. The analytic solutions of resultant equations are obtained via perturbation method. The effects of viscous dissipation and joule heating on free convection flow of a micropolar fluid with constant heat and mass fluxes have been described by Haque et al., [5]. They used Nachtsheim-Swigert iteration approach as the main tool to obtain numerical solutions. Srinivasacharya and Upendar [6,7] studied the micropolar fluid Corresponding author. E-mail address: hamzahtahak@yahoo.com (Hamzeh Taha Alkasasbeh) 55

heat and mass transfer flow over a vertical plate with out and with chemical reaction, respectively. Sheikholeslami et al., [8] discussed the mass transfer in a micropolar fluid flow through a porous channel with stationary walls by Differential Transformation Method. Fakour et al., [9] also analyzed the heat and mass transfer flow of a micropolar fluid in a channel with permeable walls. They also compared their analytic results with numerical results obtained by Runge Kutta fourth-order method. Hussanan et al., [1,11] studied analytically the micropolar fluid flow subject to Newtonian heating over a vertical plate with and without taking into account the thermal radiation effects, respectively. Recently, viscoelasticity based micropolar fluid over a stretching sheet with slip condition is considered by Sui et al., [12]. Swalmeh et al., [13] study the numerical solution free convection boundary layer flow over a solid sphere in micropolar nanofluid with prescribed wall temperature using Keller-Box method. Among the class of several other non-newtonian fluid models, Casson fluids in the presence of heat transfer is widely used in the processing of chocolate, foams, syrups, nail, toffee and many other foodstuffs [14]. Casson [15], in his pioneering work introduced this model to simulate industrial inks. Later on, a substantial study has been done on the Casson fluid flow because of its important engineering applications. Mustafa et al., [16] have studied the heat transfer flow of a Casson fluid over an impulsive motion of the plate using the homotopy method. The exact solution of forced convection boundary layer Casson fluid flow toward a linearly stretching surface with transpiration effects are reported by Mukhopadhyay et al., [17]. In the same year, Subba et al., [18] considered the velocity and thermal slip conditions on the laminar boundary layer heat transfer flow of a Casson fluid past a vertical plate. Mahdy and Ahmed [19] studied the effect of magnetohydrodynamic on a mixed convection boundary flow of an incompressible Casson fluid in the stagnation point of an impulsively rotating sphere. The convective boundary layer flow of Casson nanofluid from an isothermal sphere surface is presented by Nagendra et al., [2]. Mehmood et al., [21] investigated the micropolar Casson fluid on mixed convection flow induced by a stretching sheet. Shehzad et al [22] discussed the viscous chemical reaction effects on the MHD flow of a Casson fluid over a porous stretching sheet. Recently, Khalid et al., [23] developed exact solutions for unsteady MHD free convection flow of a Casson fluid past an oscillating plate. Amongst the various investigations on Casson fluid, the reader is referred to some new attempts made in Qasim and Noreen [24], Hussanan et al., [25], Haq et al., [26], Manjunatha, and Choudhary [27], Aman et al., [28], and Alkasasbeh [29], and the references therein. On the other hand, the flow and heat transfer phenomena about solid sphere have many important topic due to numerous engineering and industrial applications such as solving the cooling problems in turbine blades, manufacturing processes and electronic systems [3]. The exact analysis of the laminar free convection from a sphere by considering prescribed surface temperature and surface heat flux was first investigated by Chiang et al., [31]. Nazar et al., [32,33], Salleh et al., [34] and Alkasasbeh et al., [35] studied the free convection boundary layer flow on a solid sphere in a micropolar fluid with constant wall temperature, constant surface heat flux, Newtonian heating and convective boundary conditions, respectively. Based on the above contribution, the aim of present study is to investigate the effect of MHD on free convective boundary layer flow about a solid sphere in a micropolar Casson fluid and this problem has to the author knowledge not appeared thus far in the scientific literature. 2. Mathematical Analysis Consider a heated sphere of radius a, which is immersed in incompressible micropolar Casson fluid of ambient temperature. It is assumed that the surface temperature of the sphere is 56

where >. The coordinates and are chosen such that measures the distance along the surface of the sphere from the lower stagnation point and measures the distance normal to the surface of the sphere. The constitutive relationship for an incompressible Casson fluid flow, reported by Mukhopadhyay et al., [17]. = 2 + 2 >, 2 + 2 <, where =, is the (,) -th component of the deformation rate, is the plastic dynamic viscosity of the non-newtonian fluid, is a critical value of this product based on the non-newtonian model and is the yield stress of the fluid. Introducing the boundary layer approximations, the continuity, momentum, microrotation and energy equations, respectively can be written as follows. (!")+ (!#)=, (1) " % +# % = (&'() *1+, % ) -./ +1( )34* / 5.+( ) 67 8/ ", (2) ) " 67 +# 67 = 9 ) / 67 ( *2: 7 + %., (3) / ) " ; +# ; =< / ; /. (4) These equations are subjected to the boundary conditions [32], " =# =, =, :7 =, > % as =, ",,:,as, (5) where " and # are the velocity components along the and directions, respectively, :7 is the angular velocity of micropolar fluid, @ is the vortex viscosity, T is the local temperature, g is the gravity acceleration, A is the thermal conductivity, B is the electric conductivity,< is the thermal diffusivity, 1 is the thermal expansion coefficient, 1 > C is magnetic field strength, D is the kinematic viscosity, is the dynamic viscosity, E is the fluid density, =F > / H! is the microinertia density and I = 2 / is the parameter of the Casson fluid. Let!()=F34(/F)be the radial distance from the symmetrical axis to the surface of the sphere and we assume that the spin gradient viscosity J are given by J =( + @/2). (6) We introduce now the following non-dimensional variables [32], = 5, = N LM 5,! = M 5, 57

" = 5 ", # = 5 N #,: = 5/ N :7, O LM O LM O LM P Q = ;R; ; S R;, (7) where H! =1( )F T /D > is the Grashof number. Substituting variables (7) into equations (1) (4), we obtain the following non-dimensional equations of the problem under consideration. (!")+ (!#)=, (8) " % +#% =(1+U+, - )/ % /+Q34 V"+U6, (9) " 6 +#6 % = U*2:+.+*1+W 6 >./ /, (1) " X +#X =, YM / X /, (11) where U =@/ is the material or micropolar parameter, Z! =D/< is the Prandtl number and V = B1 C > F > /DE H! is the magnetic parameter. The boundary conditions (5) become " =# =, Q =1, : =, > % at =, ", Q, : as. (12) To solve equations (8) to (11), subjected to the boundary conditions (12), we assume the following variables. [ =!()\(,),Q=Q(,),: =h(,), (13) where ψ is the stream function defined as ^ ^ " =, and # M =,, (14) M which satisfies the continuity equation (1). Thus, (11) to (13) become (1+U+, _ - )P P+(1+`ab)\ / _ / *_.> + cd Q V _ +Ue =*_ *1+ W e >./ /+(1+`ab)\e h U*2h+/, / X YM /+(1+`ab)\X =*_ subject to the boundary conditions X _ X /.=*_ e _ e / _ _ / _ /., (15)., (16)., (17) 58

\ = _ =,Q =1, h =, > _ / _ /at y =,,Q,h as. (18) It can be seen that at the lower stagnation point of the sphere,, equations (15) to (17) reduce to the following nonlinear system of ordinary differential equations. (1+U+, - )\ +2\\ \ > +Q V\ +Uh =, (19) *1+ W >.h +2\h \ h U(2h+\ )=, (2), YM Q +2\Q =. (21) The boundary conditions (18) become \()=\ ()=, Q()=1,h()=, > \ (), \, Q,h as, (22) where primes denote differentiation with respect to y. The physical quantities of interest in this problem are the local skin friction coefficient g _ and the Nusselt number h %,and they can be written as g _ = 5/ where N LM P &O,h % = =k+ ( > + Y l N LM o* % >m n 5 i(; S R; ) j, (23). pc, j = A* ;.. (24) pc Using the non-dimensional variables (7) and the boundary conditions (12) the local skin friction coefficient g _ and the local Nusselt number h % are g _ =*1+ W +, _ > -.*/ /., h % = * X.. (25) pq pq 3. Graphical Results and Discussion The numerical solutions of the nonlinear system of partial differential equations (15) to (17) with boundary conditions (18) are solved by the Keller-box method (KBM) with four parameters considered, namely the Prandtl number Pr, the magnetic parameter M, the micropolar parameter K and Casson parameter I. This method is an implicit finite-difference method in conjunction with Newton s method for linearization. This is a suitable method to solve parabolic partial differential 59

equations. The boundary layer thickness =14 and step size s =.2,s =.5 are used in obtaining the numerical results. The numerical solutions start at the lower stagnation point of the sphere, with initial profiles as given by equations (19) to (21) and proceed round the sphere up to =12 q. In order to verify the accuracy of the present applied numerical scheme, a comparison with previously published results has been made. It is noticed from Table 1 that when Pr = 7, K =, 2, M = and I, the results under consideration for local Nusselt number h % reduce to the results reported by Huang and Chen [36] and Nazar et al., [32] for the case of viscous and micropolar fluids respectively. It is found that the results are a good agreement. Furthermore I believe that Keller-box method is proven to be very efficient to solve this problem motion significantly. The behavior of magnetic parameter M on local Nusselt number h % and local skin friction g _ are seen in Figure 1 and 2. It is observed that the local Nusselt number and the local skin friction are increased with the decrease in M. This behavior is in accordance with the physical observation that the application of transverse magnetic field always results in a resistive type force also called Lorentz force. This type of resisting force tends to resist the fluid flow and thus reducing the fluid. Table 1 Comparison of numerical values for the local Nusselt number h % at Pr = 7, K =, 2, M = and I, for viscous value x with previously published results of viscous and micropolar fluid K 2 Huang and Chen [36] Nazar et al., [32] Present Huang and Chen [36] Nazar et al., [32] Present o.9581.9595.959481 -.785.78451 1 o.9559.9572.95723 -.7787.77875 2 o.9496.956.95561 -.7735.773465 3 o.9389.9397.939668 -.765.7652 4 o.9239.9243.92431 -.7532.753172 5 o.945.945.9451 -.7378.7378 6 o.8858.881.8858 -.7189.718874 7 o.8518.851.85132 -.6964.696356 8 o.8182.8168.816761 -.6699.669941 9 o.7792.7792.779178 -.6393.639298 1 o - -.73277 - -.61441 11 o - -.681584 - -.5715 12 o - -.62352 - -.53577.85.8.75.7 β = 1., K = 1, Pr = 7 M=1,3,5,7.65 N u.6.55.5.45.4.35 1 2 3 4 5 6 7 8 9 1 11 12 x-degree Fig. 1. Influence of V on local Nusselt number 6

.7.6 β = 1., K = 1, Pr = 7.5 M = 1,3,5,7.4 C f.3.2.1 1 2 3 4 5 6 7 8 9 1 11 12 x-degree Fig. 2. Influence of V on local skin friction coefficient Figure 3 and 4 illustrate the influence of the Casson parameter I on the local Nusselt number and the local skin friction, respectively. It is seen from these figures that an increasing of the Casson parameter leads to increases on the local Nusselt number and decreases on the local skin friction. Moreover, as the values of increase, the rate values of the local Nusselt number decreases and the local skin friction increases. The influence of Casson parameter on temperature, velocity and angular velocity profiles are exhibited in Figure 5-7. From figure 5 that the temperature profiles Q(,) increases as decreases the values of I. Figure 4 indicates that an increase in I tends to decrease in the velocity profiles ( _ )(,). It is true because is appeared in the shear term of the momentum equation (15) and an increase in implies a decrease in yield stress of the Casson fluid. Figure 7 shows that as Casson parameter I increases, the angular velocity profiles h(,)also increases. Physically, an increase in Casson parameter means a decrease in yield stress and increases the plastic dynamic viscosity of the fluid, which makes the momentum boundary layer thicker. This effectively slows down the fluid motion..7.65 M = 5., K = 1, Pr = 7.6 N u.55.5 β = 2,3,4.45.4.35 1 2 3 4 5 6 7 8 9 1 11 12 x-degree Fig. 3. Influence of β on local Nusselt number 61

.5.45.4.35 M = 5., K = 1, Pr = 7 β = 2,3,4 C f.3.25.2.15.1.5 1 2 3 4 5 6 7 8 9 1 11 12 x-degree Fig. 4. Influence of I on local skin friction coefficient 1.8 M = 5., K = 1, Pr = 7 θ (,y).6.4 β =.1,.25,.5, 1..2 1 2 3 4 5 6 7 8 y Fig. 5. Influence of I on the temperature profiles at the lower stagnation point.6.5 M = 5, K = 1, Pr = 7.4 ( f/ y) (,y).3.2 β =.1,.25,.5, 1.1 2 4 6 8 1 12 14 y Fig. 6. Influence of I on the velocity profiles at the lower stagnation point 62

.1.5 M = 5., K = 1, Pr = 7 -.5 -.1 β =.1,.25,.5, 1. h (,y) -.15 -.2 -.25 -.3 -.35 -.4 2 4 6 8 1 12 14 y Fig. 7. Influence of I on the angular velocity profiles at the lower stagnation point Figure 8-1 illustrate the effects of several of magnetic parameter M on temperature, velocity and angular velocity profiles. The numerical results obtained show that an increase in the magnetic parameter M the values of temperature profiles Q(,) increases but values of the velocity ( _ )(,)and the angular velocity profiles h(,)decreases. This is in accordance to the physics of the problem, since the application of a transverse magnetic field results in a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity and angular velocity. 1.9 β = 2., K = 1, Pr = 7 θ (,y).8.7.6.5.4.3.2.1 M = 1, 3, 5, 7 1 2 3 4 5 6 7 8 y Fig. 8. Influence of V on the temperature profiles at the lower stagnation point 63

.1.9 β = 2., K = 1, Pr = 7.8.7 ( f/ y) (,y).6.5.4 M = 1, 3, 5, 7.3.2.1 2 4 6 8 1 12 14 y Fig. 9. Influence of V on the velocity profiles at the lower stagnation point.2.1 β = 2., K = 1, Pr = 7 h (,y) -.1 M = 1, 3, 5, 7 -.2 -.3 -.4 2 4 6 8 1 12 y Fig. 1. Influence of V on the angular velocity profiles at the lower stagnation point 4. Conclusions In this paper we have theoretically and numerically studied the problem of the effect of MHD free convective boundary layer flow about a solid sphere in a micropolar Casson fluid. We can conclude that, to get a physically acceptable solution. When the Casson parameter increases, both values of the local Nusselt number and angular velocity profiles increase, but the values of local skin friction coefficient, temperature and velocity profiles decrease. When the magnetic parameter increases, the values of the local Nusselt number, local skin friction coefficient, velocity and angular velocity profiles decrease, but the values of the temperature profiles increase. 64

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